This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A341358 #20 Mar 14 2024 03:45:44
%S A341358 315,884,1155,26325,27027,40365,44650,55335,63248,70564,72675,85936,
%T A341358 100804,106425,116624,130815,145222,224750,244036,318250,321470,
%U A341358 421515,489645,526688,531310,569625,629145,702405,730125,820808,829521,852776,885928,933224,969969
%N A341358 Deficient numbers k > 1 such that k*p is abundant for all primes p dividing k.
%C A341358 These numbers are quite close to being abundant.
%C A341358 Note that if k is a perfect number, then k*m is abundant for all m > 1. For deficient k we cannot expect this to happen since k*p must be deficient for all primes p that are large enough. However, it seems that the smallest p making k*p deficient is big compared with k. For example, the smallest prime p such that 315*p is deficient is p = 107, and the smallest prime p such that 884*p is deficient is p = 443.
%C A341358 It's actually possible that a multiple of a term (other than the term itself) is again in this sequence. However, it seems that this rarely happens. For example, both 1329548 and 1329548 * 16619 are terms.
%H A341358 Amiram Eldar, <a href="/A341358/b341358.txt">Table of n, a(n) for n = 1..1000</a> (terms 1..303 from Jianing Song)
%e A341358 The sum of divisors of 315 = 3^2 * 5 * 7 is 624 < 315 * 2, so 315 is deficient; however, all of 315 * 3, 315 * 5 and 315 * 7 are abundant, thus 315 is a term.
%t A341358 f[p_, e_] := (p^(e + 1) - 1)/(p - 1); ff[p_, e_] := (p^(e + 2) - 1)/(p^(e + 2) - p); q[n_] := Module[{fct = FactorInteger[n], s}, s = (Times @@ f @@@ fct)/n; s < 2 && AllTrue[ff @@@ fct, # > 2/s &]]; Select[Range[2, 10^5], q] (* _Amiram Eldar_, Mar 14 2024 *)
%o A341358 (PARI) isA341358(n) = if(sigma(n) < 2*n && n > 1, my(d=factor(n), k=omega(n)); for(i=1, k, if(!isA005101(n*d[i,1]), return(0))); return(1), 0) \\ see A005101 for its program
%o A341358 (PARI) is(n) = {my(f = factor(n), p = f[, 1], e = f[, 2], s = 2/sigma(f,-1)); if(n == 1 || s <= 1, return(0)); for(i = 1, #p, if((p[i]^(e[i]+2) - 1)/(p[i]^(e[i]+2) - p[i]) < s, return(0))); 1;} \\ _Amiram Eldar_, Mar 14 2024
%Y A341358 Cf. A005101 (abundant numbers), A005100 (deficient numbers).
%K A341358 nonn
%O A341358 1,1
%A A341358 _Jianing Song_, Feb 09 2021